Shurui Liu

Publications

For Computer Science and AI related projects, please see Projects.

I am mainly interested in geometric representation theory and categorical Langlands.

Preprints

Listed in reverse chronological order.

  1. Modular Representation Theory via Crystalline D-modules. Work in progress, draft available upon request, 2026.
    Abstract

    We develop a six-functor formalism of crystalline $\mathscr{D}$-modules (in the sense of Bezrukavnikov–Mirković–Rumynin) on smooth schemes and smooth Artin stacks in characteristic $p>0$ and then construct a crystalline Deligne–Lusztig induction functor from the finite Hecke category $\mathscr{D}^{\mathrm{ren}}(B\backslash G/B)$ to modular representations of a finite group of Lie type in defining characteristic. We show that this functor factors through a categorical trace via formalism in Zhu (2025). As a main result, we prove it is fully faithful and its monodromic enhancement (passing through $\mathscr{D}^{\mathrm{mon}}(U\backslash G/U)$) yields an equivalence of categories, and hence provides a framework to understand modular representations via the trace category of the finite (monodromic) Hecke category.

  2. Higher Period Integrals and Derivatives of L-functions (joint work with Zeyu Wang). Submitted, arXiv:2504.00275, 2025.
    Abstract

    We propose a geometric framework to produce a formula relating higher period integrals to higher central derivatives of $L$-functions over function fields, extending the framework of Ben-Zvi–Sakellaridis–Venkatesh to higher derivatives. For a strongly tempered affine smooth $G$-variety $X$, we give a geometric construction of the action of $L$-observables on the geometric period integral $\int_{X}\mathbb{L}_\sigma$ of a Hecke eigensheaf $\mathbb{L}_\sigma$ on $\mathrm{Bun}_G$. By taking a suitable version of Frobenius trace of this action, we recover higher central derivatives of the $L$-function attached to the dual symplectic representation. As an application, in the Rankin–Selberg case $(\mathrm{GL}_n\times\mathrm{GL}_{n-1},\mathrm{GL}_{n-1})$, we obtain a formula for higher derivatives of the Rankin–Selberg $L$-function. This provides a conceptual generalization of the higher Gross–Zagier formula of Yun–Zhang to higher-dimensional spherical varieties.

  3. Künneth Formula for Étale Fundamental Groups in Characteristic 0. Submitted, pdf, 2025.
    Abstract

    In this article, we provide a purely algebraic proof that the étale fundamental group is invariant under base change of algebraically closed fields of characteristic 0. As a corollary, we obtain the Künneth formula for étale fundamental groups in characteristic 0. We also explain how to adapt the proof (using alterations) to apply to prime-to-$p$ fundamental groups in characteristic $p>0$. These results are known to experts, but proofs do not seem to be documented in the literature, so for convenient citation by others we are providing them in a written form here.

AI for Math

  1. Automated Conjecture Resolution with Formal Verification. Haocheng Ju*, Guoxiong Gao*, Jiedong Jiang*, Bin Wu*, Zeming Sun*, Shurui Liu*, Leheng Chen, Yutong Wang, Yuefeng Wang, Zichen Wang, Wanyi He, Peihao Wu, Liang Xiao, Ruochuan Liu, Bryan Dai, Bin Dong. [Arxiv] [Rethlas] [Archon].

* Equal contribution.